Let d be a positive integer. Can a given graph G be realized in R^d so that vertices are mapped to distinct points, two vertices being adjacent if and only if the corresponding points lie on a common line that is parallel to some axis? Graphs admitting such realizations have been studied in the literature for decades under different names. Peterson asked in [Discrete Appl. Math., 2003] about the complexity of the recognition problem. While the two-dimensional case corresponds to the class of line graphs of bipartite graphs and is well-understood, the complexity question has remained open for all higher dimensions. In this paper, we answer this question. We establish the NP-completeness of the recognition problem for any fixed dimension, even in the class of bipartite graphs. To do this, we strengthen a characterization of induced subgraphs of 3-dimensional Hamming graphs due to Klavžar and Peterin. We complement the hardness result by showing that for some important classes of perfect graphs –including chordal graphs and distance-hereditary graphs– the minimum dimension of the Euclidean space in which the graph can be realized, or the impossibility of doing so, can be determined in linear time.
@InProceedings{milanic_et_al:LIPIcs.MFCS.2017.28, author = {Milanic, Martin and Mursic, Peter and Mydlarz, Marcelo}, title = {{Induced Embeddings into Hamming Graphs}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {28:1--28:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.28}, URN = {urn:nbn:de:0030-drops-81289}, doi = {10.4230/LIPIcs.MFCS.2017.28}, annote = {Keywords: gridline graph, Hamming graph, induced embedding, NP-completeness, chordal graph} }
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